Atlas › Modular forms › Weight 2 › Level 814 › 814.2.a › 814.2.a.f

Newform orbit 814.2.a.f

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q + 3q^{2} - 2q^{3} + 3q^{4} - 7q^{5} - 2q^{6} - 4q^{7} + 3q^{8} + 3q^{9} - 7q^{10} + 3q^{11} - 2q^{12} - 10q^{13} - 4q^{14} + 3q^{16} - 3q^{17} + 3q^{18} - 14q^{19} - 7q^{20} - 14q^{21} + 3q^{22} - 2q^{23} - 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{3} + 23x^{2} + 119x - 147$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
11	$-1$
37	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = +1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = -1$ : each L-function in the orbit has odd analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	3	−2	−7	−4	3	−10	−3	−14	−2	0	−4	−3	−20	−14	11	−4	−16	−20	−4	17	28	−13	3	−10	18

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.403 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 11, 22, 37, 74, 407, 814]: [0, 1, 2, 5, 3, 11, 40, 118] (certified at ell = 2147482801)
T = [(3, 1), (5, 3)]; h of degree 29 = 1 x 29 (phi(ord eps) x dim S_k^new); 5 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(814, 2)[5]   # 814.2.a.f

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 814:2 1000